Selected expository works of Shing-Tung Yau with commentary. Volume I
S.-T. Yau is a distinguished mathematician, one of the leading figures in the field of geometric analysis. Yau was awarded the Fields Medal in 1982.NEWLINENEWLINEThe \textit{Selected Expository Works of Shing-Tung Yau with Commentary} contains two volumes, for Volume II see [Zbl 1401.01046].NEWLINENEWLINEAs L. Ji mentioned in the introductory presentation, S.-T. Yau creates and developes mathematics and shares it with others; without his contributions, his visions, writings and teachings the broad subject of geometry would not have attained its current mature state.NEWLINENEWLINEThis review shortly describes the main contributions of Yau (see also \textit{A glimpse of the mathematics of Yau}, page xv, Volume I).NEWLINENEWLINEYau proposed to use nonlinear partial differential equations in geometry (differential geometry, complex geometry and algebraic geometry). There are many deep results of Yau and his co-workers in this respect, for example the Monge-Ampère equation, being understanding through geometry; the geometric structures can be effectively constructed by nonlinear equations coupled with the linear theory of index theorems, being still the only way to construct geometric structures.NEWLINENEWLINEAnother revolutionary contribution is the construction of Calabi-Yau metrics, which lead to solutions of old open problems in algebraic geometry and contributed to the development of mathematical physics and, in particular, string theory.NEWLINENEWLINEBefore 1970, theories of differential equations and differential geometry were separately and applications of nonlinear differential equations to complex and algebraic geometry could not be imagined. By Yau's and his co-authors work on geometric analysis, geometric analysis has became the most powerful development in geometry an topology. The existence of Hermitian Yang-Mills is essential for the construction of hypersymmetric heterotic strings and has a huge impact both in algebraic and complex geometry.NEWLINENEWLINEYau's work on linear differential equations has also a big influence; the gradient estimate for harmonic functions is now a standard technique in geometric analysis and the Li-Yau's estimate for the heat equation motivated the major estimates of Ricci flow, used to prove the Poincaré conjecture.NEWLINENEWLINEA great contribution of Yau and his co-workers was the proof of the famous Einstein conjecture on the positivity of mass in general relativity, using the theory of harmonic maps.NEWLINENEWLINEOn the other hand, Yau formulated conjectures and proposed many open problems, covering a wide range of topics (among others, the list of the 120 open problems given in 1982 at Princeton or the list of 100 open problems proposed in 1992), proving once more the great vision of Yau.NEWLINENEWLINEVolume I has a short preface of Shing-Tung Yau, followed by a presentation \textit{Shing-Tung Yau: His Mathematics and Writings} by L. Ji. The volume continues with some photographs of Yau and his Curriculum Vitae.NEWLINENEWLINEAfterwards, 33 papers of Yau are presented, each article having an introductory commentary (including also some history) by S.-T. Yau, the first paper being written in French. Among these are several papers not easily accessible. The expository papers of Yau are arranged according to the years of publication.NEWLINENEWLINEIn \textbf{Volume I}, the following topics studied by Yau and his co-workers are pointed-out:NEWLINENEWLINE\smallskipNEWLINENEWLINE- Kähler-Einstein metrics on open manifolds;NEWLINENEWLINE\smallskipNEWLINENEWLINE- The classical Plateau problem and the topology of \(3\)-manifolds;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Geometric bounds on the low eigenvalues of a compact surface;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Estimates of eigenvalues of a compact Riemannian manifold;NEWLINENEWLINE\smallskipNEWLINENEWLINE- The total mass and the topology of asymptotically flat space-time;NEWLINENEWLINE\smallskipNEWLINENEWLINE- The role of partial differential equations in differential geometry;NEWLINENEWLINE\smallskipNEWLINENEWLINE- The real Monge-Ampère equation and affine flat structures;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Survey on partial differential equations in differential geometry;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Problem section;NEWLINENEWLINE\smallskipNEWLINENEWLINE- A survey on Kähler-Einstein metrics;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Compact three dimensional Kähler manifolds with zero Ricci curvature;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Inequalities between Chern numbers and singular Kähler surfaces and characterization on orbit space of discrete group of \(\mathrm{SU}(2,1)\);NEWLINENEWLINE\smallskipNEWLINENEWLINE- On Ricci flat \(3\)-fold;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Some recent developments in general relativity;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Nonlinear analysis in geometry;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Three-dimensional algebraic manifolds with \(C_1=0\) and \(\chi=-6\);NEWLINENEWLINE\smallskipNEWLINENEWLINE- Uniformization of geometric structures;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Some remarks on quasi-local mass;NEWLINENEWLINE\smallskipNEWLINENEWLINE- A review of complex differential geometry;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Open problems in geometry;NEWLINENEWLINE\smallskipNEWLINENEWLINE- On the geometry of certain superconformal field theory paradigms (Towards a quantum algebraic geometry);NEWLINENEWLINE\smallskipNEWLINENEWLINE- Upper bound for the first eigenvalue of algebraic submanifolds;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Review on Kähler-Einstein metrics in algebraic geometry;NEWLINENEWLINE\smallskipNEWLINENEWLINE- A note on the distribution of critical points of eigenfunctions;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Sobolev inequalities for measure space;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Einstein manifolds with zero Ricci curvature;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Review of geometry and analysis;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Geometry and spacetime;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Geometry motivated by physics;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Complex geometry; its brief history and its future;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Geometric aspects of the moduli space of Riemannian surfaces;NEWLINENEWLINE\smallskipNEWLINENEWLINE- Solution of filtering problem with nonlinear observations;NEWLINENEWLINE\smallskipNEWLINENEWLINE- The proof of the Poincaré conjecture.NEWLINENEWLINESee the second part of this review in Volume II [Zbl 1401.01046].
scientific article; zbMATH DE number 6359373